Vacuum currents in elliptic pseudosphere tubes

TL;DR

This work computes the vacuum current density of a charged scalar in (2+1) dimensions on a background formed by an elliptic pseudosphere with a compact dimension threaded by magnetic flux. Using an exact mode-sum approach and Hadamard function decomposition, the authors isolate the topological contribution and derive two equivalent expressions for the current along the compact direction, showing it is periodic in the enclosed flux with the flux- quantum period. In the flat limit the result reduces to the known cone geometry, curvature effects are weak near the origin, and the current decays exponentially with proper distance at large separations; conformal relations further yield the currents for conformally coupled massless fields in static and hyperbolic de Sitter vacua. These results provide a precise, topology-driven description of vacuum currents in curved 2+1D geometries, with relevance to analogue gravity and condensed-matter systems exhibiting Aharonov-Bohm-type flux threading.

Abstract

We examine the effects of spatial topology, curvature, and magnetic flux on the vacuum expectation value (VEV) of the current density for a charged scalar field in (2+1)-dimensional spacetime. The elliptic pseudosphere is considered as an exactly solvable background geometry. The topological contribution is separated in the Hadamard function for general phases in the periodicity condition along the compact dimension. Two equivalent expressions are provided for the component of the current density in that direction. The corresponding VEV is a periodic function of the magnetic flux with a period equal to the flux quantum. In the flat spacetime limit, we recover the result for a conical space with a general value of the planar angle deficit. Near the origin of the elliptic pseudosphere, the effect of the spatial curvature on the vacuum current density is weak. The same applies for small values of the length of the compact dimension. Using the conformal relations between the elliptic pseudosphere and the (2+1)-dimensional de Sitter spacetime with a planar angle deficit, we determine the current densities for a conformally coupled massless scalar field in the static and hyperbolic vacuum states of locally de Sitter spacetime.

Figures (4)

• Figure 1: Elliptic pseudosphere embedded in 3D Euclidean space.
• Figure 2: The dependence of the current density on the phase $\alpha _{p}$ for different values of $\nu _{m}$ (numbers near the curves). The graphs are plotted for $R=2$ and $L/a=0.5$.
• Figure 3: The current density as a function of the coordinate $R$ for a scalar field with $\nu _{m}=0$ (left panel) and versus the parameter $\nu _{m}$ for $R=2$ (right panel). The graphs are plotted for $\alpha _{p}=\pi /2$ and the numbers near the curves present the values of the ratio $L/a$.
• Figure 4: The current density versus the ratio $L/a$ for different values of $\nu _{m}$ (numbers near the curves). The graphs are plotted for $R=2$ and $\alpha _{p}=\pi /2$.